Heroes are typically portrayed as coming from average beginnings and being relatively anonymous before the plot happens. That, or they're specifically shown to be poor, downtrodden or outcasts.
So what better way to distinguish The Rival right off the bat than by having them be just the opposite? Depending on genre, they can be anywhere from royalty/nobility, to having otherwise noteworthy parents, to just plain being loaded—with all the advantages that implies. Of course, this also makes said rival more of a challenge for the hero, which in turn makes the victory that much sweeter.

More common when protagonists are younger, because what your parents do reflects more on your personal story at that point. This trope also allows the rival to start off privileged and grow from looking down on to respecting the hero as he climbs up the ladder himself.

Examples

Anime and Manga

Naruto: Sasuke, while also an orphan, is from the Uchiha Clan, and is noteworthy for his bloodline. His father, Fugaku Uchiha was also chief of police and head of the Uchiha clan before his death. Naruto on the other hand knows nothing of his parentage and is looked down upon by everyone.

This relationship gets a lot more complicated in Shippudden however. Sasuke has pulled a Face-Heel Turn and become a most wanted criminal with only two friends and Naruto himself still loyal to him, whereas Naruto has slowly but surely gained the appreciation of the Konoha villagers and the greater Ninja world. (And it turned out his parents were actually heroes of the village and his father the Fourth Hokage. Had they not died on the day he was born, he would've be raised as a 'Prince' of the village like Tsunade or Konohamaru.)

Berserk: While Griffith is not born a Noble he eventually becomes this to Guts.

Dragon Ball Z: Vegeta is the Prince of all Saiyans, while Goku is the lineage of a low class soldier, or a circus clown.

This applies at the end of the series, where they become Vitriolic Best Buds: Vegeta lives with Bulma at Capsule Corp., while Goku lives in the mountains as a farmer.

Pokémon: Gary Oak, being how his grandfather gives out starting Pokemon, gets first choice of starters and spent his journey through both Kanto and neighboring Johto practically walking on air, unlike his rival, Ash, who was late and got stuck with an unruly Pikachu, and struggled to just get enough badges to qualify for the Indigo tournament (even though Ash goes farther than Gary, whose elimination happens off-screen). Ash absolved his rivalry with Gary after defeating him at the climax of the Johto saga, which led to Gary stepping out of competitive battling altogether.

Gundam Wing: Zechs is later revealed to be the missing heir of the Peacecraft name.

Vali Lucifer in High School DXD is this for Issei. The former is a descendant of the original Lucifer so that gives him a high amount of magical reserves, is capable of using Juggernaut Drive without any defects whatsoever for a few minutes, and is capable of learning spells just by reading them. The latter however comes from an ordinary background with normal parents, and just used Boosted Gear for only six months. They also have different tastes in their fetish where the former loves the female buttocks, while the latter loves female breasts.

Ashita no Joe : Toru Rikiishi is this to protagonist Joe Yabuki. The former is more collected and cultured and a better-functioning member of society, with much more experience in professional boxing. Joe is an orphan who has spent his entire life in the streets, is an absolute jerk and his bratty and impulsive behavior induces fear and irritation whereas Rikiishi's induces admiration and respect. Besides, Joe trains in a hovel under a bridge while Rikiishi has a gym built just for him by a rich family. They met when in juvenile prison but while Joe got there due to his misbehavior, Rikiishi was injustly imprisoned (if not downright framed).

Kiyosumi are a bunch of nobodies, while Kazekoshi has long and illustrious history in competitive Mahjong; on the other hand Ryuumonbuchi is loaded with fat stacks of cash and "buy" talents from all over the country (literally, in case of Hajime). Guess who made it to National Championship?

It gets more extreme in the spin-off Saki Achiga-hen: Achiga is a bunch of nobodies and their captain is a monkey. They are expected to go against Shiratodai, who are monsters among monsters and whose leader is the arch-nemesis of the main series' protagonists.

Girls und Panzer (which can be dubbed as Saki on tanks) has a school that is without Tankery up until the last year, and is expected to go against: St. Gloriana, who is proud of their history in Tankery; Saunders, who deploys lots and lots of tanks; and Black Peak, who has the legendary Nishizumi Maho and her elite panzerkorps.

Played with in Captain Tsubasa. Tsubasa's fist rival Genzo Wakabayashi is a Princely Young Man who plays for a private school's team which is full of talented rich kids, so he plays it straight until Defeat Means Friendship. Tsubasa's second rival, however, is Kojiro Hyuuga, who is very underprivileged since he comes from a very poor family and thus inverts the trope. (He tries to shame Tsubasa for being from a middle-to-high class family, but fails.) Later it's zigzagged with Jun Misugi, another super talented Princely Young Man... who's actually an Ill Boy and thus Tsubasa has a huge advantage over him.

Genzo ends up having his own Privileged Rival in Karl-Heinz Schneider, while his and Tsubasa's friend Misaki has one in El-Si Pierre. In both cases the boys have left Japan to prove themselves in Germany and France, respectively, and in Schneider's he also was the first rival who also offered Genzo respect after he was bullied for being a foreigner.

Gattaca: Since the film takes place in a world where people where having good genes means success, Anton is this to Vincent despite them being raised by the same family.

Literature

Deryni Rising has Kelson (young, inexperienced, labouring under a religious and social system that persecutes Deryni) facing Charissa, who's about a decade older, highly trained in arcana, wealthy and related to the royal family of the neighbouring kingdom that has no such history of persecution.

Harry Potter: Harry's rival Draco Malfoy is from another powerful magic family. His father is shown to have the ear of the Minster for Magic Cornelius Fudge.

A Wizard of Earthsea. When Ged arrives at Roke Island, he gains a rival named Jasper, who's the son of the Lord of the Domain of Eolg on the Isle of Havnor. Ged is just the son of a smith, and is rubbed the wrong way by Jasper's extremely polite but condescending manners.

Live-Action Television

Veronica Mars: Most of the villains are villainous because they have the money to bribe, cheat, and scam their way through the court system which effectively kneecaps the only thing Veronica can do (sleuth, get evidence, get them in court).

Parks and Recreation gives us Leslie's opponent to the City Council seat, uber-privileged and oblivious Bobby Newport, heir to the Sweetums empire.

On a wider scale, there's the town rivalry between Pawnee and its more upscale neighbor Eagleton.

In Round the Twist, the Twist kids' nemesis is the rich kid James Gribble, whose dad eventually ends up mayor of their town.

Tekken has Asuka Kazama and Emilie "Lili" de Rochefort, who can be seen as Alternate Company Equivalents of the aforementioned Sakura and Karin. Asuka is a scrappy street fighter and ordinary schoolgirl; Lili is the daughter of a wealthy oil magnate from Monaco. Lili loses to Asuka in 5: Dark Resurrection and, driven by revenge, spends the next tournament trying to challenge Asuka to a rematch. In her ending, Lili's butler Sebastian comments that fighting is how Lili makes friends, so make of that what you will.

The Fairly OddParents: Timmy's only rival that also has a Fairy God Parent is named Remy Buxaplenty for obvious reasons.

Avatar: The Last Airbender: Zuko, prince of the Fire Nation is this to Aang, an otherwise anonymous monk before it's revealed that he's the Avatar.

Amusingly, Zuko also has Azula as his Privileged Rival. Azula has the love from their father that Zuko never did. In Season 2 we have the contrast of now penniless Zuko, hunted as an enemy of the Earth Kingdom and traitor to the Fire Nation vs Azula, who has all the wealth, resources, and influence of the Fire Nation behind her. Both are actively chasing Aang, but decidedly not working together.

The Smythe-Higgins family from Hey Arnold!, Arnold's family's intergenerational rivals, are shown to be of the upper-class.

Spongebob Squarepants: Squidwards's rival is Squilliam who 'got rich doing everything Squidward wanted to do' while Squidward is just a cashier at a fastfood joint.

My Little Pony: Friendship Is Magic: Diamond Tiara and Silver Spoon are blatantly this to the Cutie Mark Crusaders. As if it isn't bad enough having this Spoiled Brat duo incessantly mock CMC for their "blank flanks", they are apparently well off and also seem to look down on CMC for supposedly being lower class, a notion not unjustified looking at their backgrounds:

When Sweetie Belle's parents made an appearance, they came off as typical middle, if not lower, class ruralites. Meanwhile, Diamond's father is a business pony aptly named Filthy Rich. While Sweetie's sister Rarity has established herself as an equally successful business pony, Mr. Rich has obviously been doing it longer and is therefore more established in society.

We don't even know if Scootaloo has a family. While the same could be said for Silver, at least she appears to be taken care of while anything is possible for rough and tumble tomboy Scoot. Season 4 at least revealed that she has a proper dwelling, yet mum is still the word about her family.

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